set.seed(1998)
x <- rnorm(n = 3000, mean = 0.01, sd = 0.02)
# plot(x)
# plot(cumsum(x))
dd <- data.frame(cbind("k" = 1:length(x), "x_t" = x))
ggplot(dd) +
geom_line(aes(x = k, y = x_t), size = 0.35)
dd <- data.frame(cbind("k" = 1:length(x), "y_t" = cumsum(x)))
ggplot(dd) +
geom_line(aes(x = k, y = y_t), size = 1, colour = "gold3")
n <- 3000
nu <- 10
mu <- 0.001
vol <- 0.005*2
scaler <- sqrt(vol^2/(nu)*(nu - 2))
set.seed(1998)
x <- mu + scaler*rt(n = n, df = nu)
x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
y <- cumsum(x)
dd <- data.frame(cbind("k" = 1:length(x), "x_t" = x))
ggplot(dd) +
geom_line(aes(x = k, y = x_t), size = 0.35)
dAll <- seq(from = 0, to = 1, length.out = 11)
fracDiffAll <- xts()
for (i in 1:length(dAll)) {
d <- dAll[i]
th <- 1e-4
yDiff <- na.omit(fracDiff_FFD(y, d, th))
fracDiffAll <- cbind(fracDiffAll, yDiff)
}
colnames(fracDiffAll) <- paste("d =", as.character(dAll))
index(fracDiffAll) <- index(y)
plotTimeSeries(fracDiffAll)
allD <- seq(from = 0, to = 1, length.out = 21)
allADFStatistics <- rep(NA, length(allD))
stationaryTSeries<- rep(FALSE, length(allD))
for (i in 1:length(allD)) {
tSeriesFracDiff_FFD <- na.omit(fracDiff_FFD(y, allD[i], 1e-4))
ADFTest <- adf.test(tSeriesFracDiff_FFD)
allADFStatistics[i] <- ADFTest$statistic
if (ADFTest$p.value <= 0.01) stationaryTSeries[i] <- TRUE
}
dataADF <- data.frame(cbind(allD, allADFStatistics, as.factor(stationaryTSeries)))
ggplot(dataADF, aes(x = allD, y = allADFStatistics)) +
geom_line(colour = "gold2", size = 1) +
geom_point(aes(fill = stationaryTSeries), size = 3, shape = 21, stroke = 0) +
scale_fill_manual(values = c("red2", "green3")) +
xlab("d") + ylab("ADF Statistic") +
scale_x_continuous(breaks = seq(from = 0, to = 1, length.out = 11)) +
ggtitle("ADF Test Statistic as a function of d")
IID-Samples
d <- 0.35
# d <- 0.15
th <- 1e-4
nFwd <- 150 # 7 months
set.seed(1998)
totMeanDiff <- totMeanDiffLM <- totMeanRet <- 0
# allwInverse <- list()
# for (i in 1:9) {
# print(i)
# allwInverse[[i]] <- getInverseWeights_FFD(i/10, th, n + nFwd)
# }
#
# saveRDS(allwInverse, file = "allwInverse.Rds")
wInverse <- getInverseWeights_FFD(d, th, n + nFwd)
nIter <- 50
for (i in 1:nIter) {
# print(paste("Iteracion", i))
x <- mu + scaler*rt(n = n, df = nu)
x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
y <- cumsum(x)
# y <- c(0, 0, 0)
# for (k in 4:(n + 1)) {
# y <- c(y, 0.5 + 0.3*y[k - 1] + 0.4*y[k - 2] + 0.297*y[k - 3]
# + rnorm(1, mean = 0, sd = 1.5))
# }
# x <- diff(y)
# x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
# y <- as.xts(y[-1], order.by = index(x))
#
# print(adf.test(y))
# plot(y)
# x <- mu + scaler*rt(n = n, df = nu)
# for (i in 1:length(x)) {
# if (mod(i, 100) == 35) {
# x[i:(i + 4)] <- rnorm(5, mean = -30*mu, sd = 0.006)
# x[(i + 5):(i + 14)] <- rnorm(10, mean = 0, sd = 0.003)
# x[(i + 15):(i + 24)] <- rnorm(10, mean = 15*mu, sd = 0.003)
# }
# }
# x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
# y <- cumsum(x)
yTrn <- y[1:(length(y) - nFwd)]
yTst <- y[(length(y) - nFwd + 1):length(y)]
yDiffOriginal <- fracDiff_FFD(yTrn, d, th)
yDiff <- completeFracDiff_FFD(yTrn, yDiffOriginal, d, th)
# _____ Linear Model _____
yLM <- as.vector(yDiff[(length(yDiff) - 21*6*5):length(yDiff)])
xLM <- 1:length(yLM)
dd <- data.frame(xLM, yLM)
linearMod <- lm(yLM~xLM, data = dd)
dd <- data.frame("xLM" = (length(xLM) + 1):(length(xLM) + nFwd))
yDiffLM <- c(yDiff, as.xts(predict(linearMod, dd),
order.by = tail(index(yDiff), 1) + 1:nFwd))
yIntLM <- inverseFracDiff_FFD(yDiffLM, d, th, w = wInverse)
# _______________
muDiff <- mean(yDiff[(length(yDiff) - 21*6*5):length(yDiff)])
yDiff <- c(yDiff, as.xts(rep(muDiff, nFwd),
order.by = tail(index(yDiff), 1) + 1:nFwd))
yDiffOriginal <- c(yDiffOriginal,
as.xts(rep(muDiff, nFwd),
order.by = tail(index(yDiffOriginal), 1) + 1:nFwd))
yInt <- inverseFracDiff_FFD(yDiff, d, th, w = wInverse)
yDiff2 <- fracDiff(yTrn, 1, 1)
muDiff2 <- mean(yDiff2[(length(yDiff2) - 21*6*5):length(yDiff2)])
yDiff2 <- c(yDiff2, as.xts(rep(muDiff2, nFwd),
order.by = tail(index(yDiff2), 1) + 1:nFwd))
yInt2 <- cumsum(yDiff2)
dd <- cbind("y" = y, "yDiff" = yDiffOriginal, "yInt" = yInt, "yIntLM" = yIntLM,
"yDiff2" = yDiff2, "yInt2" = yInt2)
if (i <= 5) print(plotTimeSeries(dd))
dd <- cbind("y" = yTst, "yInt" = yInt[index(yTst)],
"yIntLM" = yIntLM[index(yTst)],
"yInt2" = yInt2[index(yTst)])
if (i <= 5) print(plotTimeSeries(dd))
totMeanDiff <- 1/length(yTst)*sum(abs(yTst - yInt[index(yTst)])^2) + totMeanDiff
totMeanDiffLM <- 1/length(yTst)*sum(abs(yTst - yIntLM[index(yTst)])^2) + totMeanDiffLM
totMeanRet <- 1/length(yTst)*sum(abs(yTst - yInt2[index(yTst)])^2) + totMeanRet
}
totMeanDiff <- totMeanDiff/nIter
totMeanDiffLM <- totMeanDiffLM/nIter
totMeanRet <- totMeanRet/nIter
print(totMeanDiff)
## [1] 0.05301019
print(totMeanDiffLM)
## [1] 0.01258059
print(totMeanRet)
## [1] 0.007291666
Non-IID Samples (Ver. 1)
# d <- 0.35
d <- 0.15
th <- 1e-4
nFwd <- 150 # 7 months
set.seed(1995)
totMeanDiff <- totMeanDiffLM <- totMeanRet <- 0
wInverse <- getInverseWeights_FFD(d, th, n + nFwd)
x <- mu + scaler*rt(n = n, df = nu)
for (i in 1:length(x)) {
if (mod(i, 100) == 35) {
x[i:(i + 4)] <- rnorm(5, mean = -30*mu, sd = 0.006)
x[(i + 5):(i + 14)] <- rnorm(10, mean = 0, sd = 0.003)
x[(i + 15):(i + 24)] <- rnorm(10, mean = 15*mu, sd = 0.003)
}
}
x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
y <- cumsum(x)
plot(x[1:100])
print(adf.test(y))
##
## Augmented Dickey-Fuller Test
##
## data: y
## Dickey-Fuller = -3.7585, Lag order = 14, p-value = 0.02105
## alternative hypothesis: stationary
stationaryTSeries[] <- FALSE
for (i in 1:length(allD)) {
tSeriesFracDiff_FFD <- na.omit(fracDiff_FFD(y, allD[i], 1e-4))
ADFTest <- adf.test(tSeriesFracDiff_FFD)
allADFStatistics[i] <- ADFTest$statistic
if (ADFTest$p.value <= 0.01) stationaryTSeries[i] <- TRUE
}
dataADF <- data.frame(cbind(allD, allADFStatistics, as.factor(stationaryTSeries)))
print(ggplot(dataADF, aes(x = allD, y = allADFStatistics)) +
geom_line(colour = "gold2", size = 1) +
geom_point(aes(fill = stationaryTSeries), size = 3, shape = 21, stroke = 0) +
scale_fill_manual(values = c("red2", "green3")) +
xlab("d") + ylab("ADF Statistic") +
scale_x_continuous(breaks = seq(from = 0, to = 1, length.out = 11)) +
ggtitle("ADF Test Statistic as a function of d"))
nIter <- 50
for (i in 1:nIter) {
x <- mu + scaler*rt(n = n, df = nu)
for (k0 in 1:length(x)) {
if (mod(k0, 100) == 35) {
x[k0:(k0 + 4)] <- rnorm(5, mean = -30*mu, sd = 0.006)
x[(k0 + 5):(k0 + 14)] <- rnorm(10, mean = 0, sd = 0.003)
x[(k0 + 15):(k0 + 24)] <- rnorm(10, mean = 15*mu, sd = 0.003)
}
}
x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
y <- cumsum(x)
yTrn <- y[1:(length(y) - nFwd)]
yTst <- y[(length(y) - nFwd + 1):length(y)]
yDiffOriginal <- fracDiff_FFD(yTrn, d, th)
yDiff <- completeFracDiff_FFD(yTrn, yDiffOriginal, d, th)
# _____ Linear Model _____
yLM <- as.vector(yDiff[(length(yDiff) - 21*6*5):length(yDiff)])
xLM <- 1:length(yLM)
dd <- data.frame(xLM, yLM)
linearMod <- lm(yLM~xLM, data = dd)
dd <- data.frame("xLM" = (length(xLM) + 1):(length(xLM) + nFwd))
yDiffLM <- c(yDiff, as.xts(predict(linearMod, dd),
order.by = tail(index(yDiff), 1) + 1:nFwd))
yIntLM <- inverseFracDiff_FFD(yDiffLM, d, th, w = wInverse)
# _______________
muDiff <- mean(yDiff[(length(yDiff) - 21*6*5):length(yDiff)])
yDiff <- c(yDiff, as.xts(rep(muDiff, nFwd),
order.by = tail(index(yDiff), 1) + 1:nFwd))
yDiffOriginal <- c(yDiffOriginal,
as.xts(rep(muDiff, nFwd),
order.by = tail(index(yDiffOriginal), 1) + 1:nFwd))
yInt <- inverseFracDiff_FFD(yDiff, d, th, w = wInverse)
yDiff2 <- fracDiff(yTrn, 1, 1)
muDiff2 <- mean(yDiff2[(length(yDiff2) - 21*6*5):length(yDiff2)])
yDiff2 <- c(yDiff2, as.xts(rep(muDiff2, nFwd),
order.by = tail(index(yDiff2), 1) + 1:nFwd))
yInt2 <- cumsum(yDiff2)
dd <- cbind("y" = y, "yDiff" = yDiffOriginal, "yInt" = yInt, "yIntLM" = yIntLM,
"yDiff2" = yDiff2, "yInt2" = yInt2)
if (i <= 5) print(plotTimeSeries(dd))
dd <- cbind("y" = yTst, "yInt" = yInt[index(yTst)],
"yIntLM" = yIntLM[index(yTst)],
"yInt2" = yInt2[index(yTst)])
if (i <= 5) print(plotTimeSeries(dd))
totMeanDiff <- 1/length(yTst)*sum(abs(yTst - yInt[index(yTst)])^2) + totMeanDiff
totMeanDiffLM <- 1/length(yTst)*sum(abs(yTst - yIntLM[index(yTst)])^2) + totMeanDiffLM
totMeanRet <- 1/length(yTst)*sum(abs(yTst - yInt2[index(yTst)])^2) + totMeanRet
}
totMeanDiff <- totMeanDiff/nIter
totMeanDiffLM <- totMeanDiffLM/nIter
totMeanRet <- totMeanRet/nIter
print(totMeanDiff)
## [1] 0.06859932
print(totMeanDiffLM)
## [1] 0.01691159
print(totMeanRet)
## [1] 0.02652752
Non-IID Samples (Ver. 2)
n <- 3000
mu <- 0.01
vol <- 0.03
period <- 21*6 # 6 months
muSin <- mu*sin(mod(1:n, period)/period*2*pi)
set.seed(1995)
x <- c()
for (i in 1:length(muSin)) {
x <- c(x, rnorm(n = 1, mean = muSin[i], sd = vol))
}
x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
y <- cumsum(x)
dd <- data.frame(cbind("k" = 1:length(x), "x_t" = x))
ggplot(dd) +
geom_line(aes(x = k, y = x_t), size = 0.35)
dAll <- seq(from = 0, to = 1, length.out = 11)
fracDiffAll <- xts()
for (i in 1:length(dAll)) {
d <- dAll[i]
th <- 1e-4
yDiff <- na.omit(fracDiff_FFD(y, d, th))
print(adf.test(yDiff))
fracDiffAll <- cbind(fracDiffAll, yDiff)
}
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -3.8284, Lag order = 14, p-value = 0.01756
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -4.316, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -4.6916, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -5.1811, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -5.8904, Lag order = 13, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -6.4821, Lag order = 14, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -7.1621, Lag order = 14, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -7.8468, Lag order = 14, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -8.7142, Lag order = 14, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -9.7792, Lag order = 14, p-value = 0.01
## alternative hypothesis: stationary
##
##
## Augmented Dickey-Fuller Test
##
## data: yDiff
## Dickey-Fuller = -10.914, Lag order = 14, p-value = 0.01
## alternative hypothesis: stationary
colnames(fracDiffAll) <- paste("d =", as.character(dAll))
index(fracDiffAll) <- index(y)
plotTimeSeries(fracDiffAll)
d <- 0.2
th <- 1e-4
nFwd <- 150 # 1 month
set.seed(1998)
wInverse <- getInverseWeights_FFD(d, th, length(y) + nFwd)
x <- c()
for (i in 1:length(muSin)) {
x <- c(x, rnorm(n = 1, mean = muSin[i], sd = vol))
}
x <- as.xts(x, order.by = as.Date("2000-01-01") + 1:n)
y <- cumsum(x)
yTrn <- y[1:(length(y) - nFwd)]
yTst <- y[(length(y) - nFwd + 1):length(y)]
yDiffOriginal <- fracDiff_FFD(yTrn, d, th)
yDiff <- completeFracDiff_FFD(yTrn, yDiffOriginal, d, th)
# _____ Linear Model _____
yLM <- as.vector(yDiff[(length(yDiff) - 21*6):length(yDiff)])
xLM <- 1:length(yLM)
dd <- data.frame(xLM, yLM)
linearMod <- lm(yLM~xLM, data = dd)
dd <- data.frame("xLM" = (length(xLM) + 1):(length(xLM) + nFwd))
yDiffLM <- c(yDiff, as.xts(predict(linearMod, dd),
order.by = tail(index(yDiff), 1) + 1:nFwd))
yIntLM <- inverseFracDiff_FFD(yDiffLM, d, th, w = wInverse)
# _______________
muDiff <- mean(yDiff[(length(yDiff) - 21*6):length(yDiff)])
yDiff <- c(yDiff, as.xts(rep(muDiff, nFwd),
order.by = tail(index(yDiff), 1) + 1:nFwd))
yDiffOriginal <- c(yDiffOriginal,
as.xts(rep(muDiff, nFwd),
order.by = tail(index(yDiffOriginal), 1) + 1:nFwd))
yInt <- inverseFracDiff_FFD(yDiff, d, th, w = wInverse)
yDiff2 <- fracDiff(yTrn, 1, 1)
muDiff2 <- mean(yDiff2[(length(yDiff2) - 21*6):length(yDiff2)])
yDiff2 <- c(yDiff2, as.xts(rep(muDiff2, nFwd),
order.by = tail(index(yDiff2), 1) + 1:nFwd))
yInt2 <- cumsum(yDiff2)
dd <- cbind("y" = y, "yDiff" = yDiffOriginal, "yInt" = yInt, "yIntLM" = yIntLM,
"yDiff2" = yDiff2, "yInt2" = yInt2)
print(plotTimeSeries(dd))
dd <- cbind("y" = yTst, "yInt" = yInt[index(yTst)],
"yIntLM" = yIntLM[index(yTst)],
"yInt2" = yInt2[index(yTst)])
print(plotTimeSeries(dd))
print(1/length(yTst)*sum(abs(yTst - yInt[index(yTst)])^2))
## [1] 0.115185
print(1/length(yTst)*sum(abs(yTst - yIntLM[index(yTst)])^2))
## [1] 0.7207504
print(1/length(yTst)*sum(abs(yTst - yInt2[index(yTst)])^2))
## [1] 0.2107852